When you put the spell on the stack, you choose an opponent, then that opponent chooses the target. This is not a choice made on resolution.
The target of the spell is a creature the chosen opponent controls. As Evangelize resolves, if the target isn't a creature controlled by the chosen opponent, Evangelize won't resolve. If Evangelize's target is changed (via Shunt, for example), the new target must be a creature controlled by the chosen opponent.